On a Question of N. Th. Varopoulos and the constant $C_2(n)$

Gupta, Rajeev; Ray, Samya K.

doi:10.5802/aif.3218

On a Question of N. Th. Varopoulos and the constant

C_{2} (n)

[Sur une question de N. Th. Varopoulos et la constante

C_{2} (n)

]

Gupta, Rajeev ¹ ; Ray, Samya K.¹

¹ Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur-208016 (India)

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2613-2634.

Résumé
Abstract

Soit $ℂ_{k} [Z_{1}, ..., Z_{n}]$ l’ensemble de tous les polynômes de degré au plus $k$ dans $n$ variables complexes et $𝒞_{n}$ l’ensemble de tous les $n$ -tuples $T = (T_{1}, ..., T_{n})$ de contractions qui commutent sur un espace de Hilbert $ℍ .$ L’inégalité intéressante

K_{G}^{ℂ} \leq lim_{n \to \infty} C_{2} (n) \leq 2 K_{G}^{ℂ},

où

C_{k} (n) = sup \{{∥ p (T) ∥ : ∥ p ∥}_{𝔻^{n}, \infty} \leq 1, p \in ℂ_{k} [Z_{1}, ..., Z_{n}], T \in 𝒞_{n}\}

et $K_{G}^{ℂ}$ est la constante complexe de Grothendieck, est due à Varopoulos. Nous répondons à une question de longue date en montrant que ${lim}_{n \to \infty} \frac{C_{2} (n)}{K_{G}^{ℂ}}$ est strictement plus grand que $1$ . Soit $ℂ_{2}^{s} [Z_{1}, ..., Z_{n}]$ l’ensemble des polynômes homogènes complexes $p (z_{1}, ..., z_{n}) = \sum_{j, k = 1}^{n} a_{j k} z_{j} z_{k}$ de degré deux dans $n$ -variables, oú $((a_{j k}))$ est une matrice symétrique complexe $n \times n$ . Pour chaque $n \in ℕ,$ on définit la carte linéaire $𝒜_{n} : (ℂ_{2}^{s} [Z_{1}, ..., Z_{n}] {, ∥ \cdot ∥}_{𝔻^{n}, \infty}) \to (M_{n} {, ∥ \cdot ∥}_{\infty \to 1})$ par $𝒜_{n} (p) = ((a_{j k})) .$ Nous montrons que le supremum (sur $n$ ) de la norme des opérateurs $𝒜_{n}; n \in ℕ,$ est borné inférieurement par la constante $π^{2} / 8 .$ En utilisant une classe d’opérateurs, introduite par Varopoulos, nous construisons aussi une grande classe de polynômes explicites pour laquelle l’inégalité de von Neumann n’est pas satisfaite. Nous prouvons que le polynôme de Varopoulos–Kaijser est extrémal parmi une grande classe convenablement choisie de polynômes homogènes de degré deux. Nous étudions également le comportement de la constante $C_{k} (n)$ lorsque $n \to \infty .$

Let $ℂ_{k} [Z_{1}, ..., Z_{n}]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and $𝒞_{n}$ denote the set of all $n$ -tuple $T = (T_{1}, ..., T_{n})$ of commuting contractions on some Hilbert space $ℍ .$ The interesting inequality

K_{G}^{ℂ} \leq lim_{n \to \infty} C_{2} (n) \leq 2 K_{G}^{ℂ},

where

C_{k} (n) = sup \{{∥ p (T) ∥ : ∥ p ∥}_{𝔻^{n}, \infty} \leq 1, p \in ℂ_{k} [Z_{1}, ..., Z_{n}], T \in 𝒞_{n}\}

and $K_{G}^{ℂ}$ is the complex Grothendieck constant, is due to Varopoulos. We answer a long–standing question by showing that the limit ${lim}_{n \to \infty} \frac{C_{2} (n)}{K_{G}^{ℂ}}$ is strictly bigger than $1 .$ Let $ℂ_{2}^{s} [Z_{1}, ..., Z_{n}]$ denote the set of all complex valued homogeneous polynomials $p (z_{1}, ..., z_{n}) = \sum_{j, k = 1}^{n} a_{j k} z_{j} z_{k}$ of degree two in $n$ -variables, where $((a_{j k}))$ is a $n \times n$ complex symmetric matrix. For each $n \in ℕ,$ define the linear map $𝒜_{n} : (ℂ_{2}^{s} [Z_{1}, ..., Z_{n}] {, ∥ \cdot ∥}_{𝔻^{n}, \infty}) \to (M_{n} {, ∥ \cdot ∥}_{\infty \to 1})$ to be $𝒜_{n} (p) = ((a_{j k})) .$ We show that the supremum (over $n$ ) of the norm of the operators $𝒜_{n}; n \in ℕ,$ is bounded below by the constant $π^{2} / 8 .$ Using a class of operators, first introduced by Varopoulos, we also construct a large class of explicit polynomials for which the von Neumann inequality fails. We prove that the original Varopoulos–Kaijser polynomial is extremal among a, suitably chosen, large class of homogeneous polynomials of degree two. We also study the behaviour of the constant $C_{k} (n)$ as $n \to \infty .$

Reçu le : 2017-05-05
Révisé le : 2017-11-09
Accepté le : 2018-01-11
Publié le : 2018-11-23

DOI : 10.5802/aif.3218

Classification : 47A13, 47A25, 47A60, 47A63
Keywords: Grothendieck Inequality, von Neumann Inequality, Varopoulos Operator, Grothendieck Constant, Positive Grothendieck Constant
Mot clés : Inégalité de Grothendieck, inégalité de von Neumann, opérateur de Varopoulos, Constante de Grothendieck, Constante de Grothedieck positive

Affiliations des auteurs :

Gupta, Rajeev ¹ ; Ray, Samya K. ¹

¹ Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur-208016 (India)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2018__68_6_2613_0,
     author = {Gupta, Rajeev and Ray, Samya K.},
     title = {On a {Question} of {N.} {Th.} {Varopoulos} and the constant $C_2(n)$},
     journal = {Annales de l'Institut Fourier},
     pages = {2613--2634},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3218},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3218/}
}

TY  - JOUR
AU  - Gupta, Rajeev
AU  - Ray, Samya K.
TI  - On a Question of N. Th. Varopoulos and the constant $C_2(n)$
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2613
EP  - 2634
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3218/
DO  - 10.5802/aif.3218
LA  - en
ID  - AIF_2018__68_6_2613_0
ER  -

%0 Journal Article
%A Gupta, Rajeev
%A Ray, Samya K.
%T On a Question of N. Th. Varopoulos and the constant $C_2(n)$
%J Annales de l'Institut Fourier
%D 2018
%P 2613-2634
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3218/
%R 10.5802/aif.3218
%G en
%F AIF_2018__68_6_2613_0

Gupta, Rajeev; Ray, Samya K. On a Question of N. Th. Varopoulos and the constant $C_2(n)$. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2613-2634. doi : 10.5802/aif.3218. https://aif.centre-mersenne.org/articles/10.5802/aif.3218/

Bibliographie
Cité par

[1] Alon, Noga; Naor, Assaf Approximating the cut-norm via Grothendieck’s inequality, SIAM J. Comput., Volume 35 (2006) no. 4, pp. 787-803 | DOI | MR | Zbl

[2] Andô, Tadao On a pair of commutative contractions, Acta Sci. Math., Volume 24 (1963), pp. 88-90 | MR | Zbl

[3] Bagchi, Bhaskar; Misra, Gadadhar Contractive homomorphisms and tensor product norms, Integral Equations Oper. Theory, Volume 21 (1995) no. 3, pp. 255-269 | DOI | MR | Zbl

[4] Bagchi, Bhaskar; Misra, Gadadhar On Grothendieck Constants (2008) (www.math.iisc.ac.in/ gm/gmhomefiles/papers/kgc2.pdf)

[5] Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf The Grothendieck constant is strictly smaller than Krivine’s bound, Forum Math. Pi, Volume 1 (2013), 4, 42 pages (Art. ID. 4, 42 p.) | MR | Zbl

[6] Crabb, M. J.; Davie, Alexander M. von Neumann’s inequality for Hilbert space operators, Bull. Lond. Math. Soc., Volume 7 (1975), pp. 49-50 | MR | Zbl

[7] Fishburn, Peter C.; Reeds, James A. Bell inequalities, Grothendieck’s constant, and root two, SIAM J. Discrete Math., Volume 7 (1994) no. 1, pp. 48-56 | DOI | MR | Zbl

[8] Gupta, Rajeev The Carathéodory-Fejér Interpolation Problems and the von-Neumann Inequality (2015) (https://arxiv.org/abs/1508.07199)

[9] Gupta, Rajeev An improvement on the bound for $C_{2} (n)$ , Acta Sci. Math., Volume 83 (2017) no. 1-2, pp. 263-269 | Zbl

[10] Holbrook, John A. Schur norms and the multivariate von Neumann inequality, Recent advances in operator theory and related topics (Szeged, 1999) (Operator Theory: Advances and Applications), Volume 127, Birkhäuser, 2001, pp. 375-386 | MR | Zbl

[11] Holbrook, John A.; Schoch, Jean-Pierre Theory vs. experiment: multiplicative inequalities for the numerical radius of commuting matrices, Topics in operator theory. Volume 1. Operators, matrices and analytic functions (Operator Theory: Advances and Applications), Volume 202, Birkhäuser, 2010, pp. 273-284 | DOI | MR | Zbl

[12] Krivine, Jean-Louis Sur la constante de Grothendieck, C. R. Acad. Sci. Paris, Sér. A, Volume 284 (1977) no. 8, pp. 445-446 | MR | Zbl

[13] Krivine, Jean-Louis Constantes de Grothendieck et fonctions de type positif sur les sphères, Adv. Math., Volume 31 (1979) no. 1, pp. 16-30 | DOI | MR | Zbl

[14] Li, Chi-Kwong; Tam, Bit-Shun A note on extreme correlation matrices, SIAM J. Matrix Anal. Appl., Volume 15 (1994) no. 3, pp. 903-908 | DOI | MR | Zbl

[15] von Neumann, Johann Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr., Volume 4 (1951), pp. 258-281 | MR | Zbl

[16] Niculescu, Constantin; Persson, Lars-Erik Convex functions and their applications. A contemporary approach, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, 2006, xvi+255 pages | DOI | MR | Zbl

[17] Pisier, Gilles Grothendieck inequality, random matrices and quantum expanders (based on SGU Special lectures at Kyoto University from March 12 to March 18, 2015, recorded by Kei Hasegawa, https://ktgu.math.kyoto-u.ac.jp/sites/default/files/Pisier_lecturenote.pdf)

[18] Pisier, Gilles Factorization of linear operators and geometry of Banach spaces, Regional Conference Series in Mathematics, 60, American Mathematical Society, 1986, x+154 pages | DOI | MR | Zbl

[19] Pisier, Gilles Similarity problems and completely bounded maps. Includes the solution to “The Halmos problem”, Lecture Notes in Mathematics, 1618, Springer, 2001, viii+198 pages | DOI | MR | Zbl

[20] Pisier, Gilles Grothendieck’s theorem, past and present, Bull. Am. Math. Soc., Volume 49 (2012) no. 2, pp. 237-323 | DOI | MR

[21] Ray, Samya Kumar On Multivariate Matsaev’s Conjecture (2017) (https://arxiv.org/abs/1703.00733)

[22] Varopoulos On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Funct. Anal., Volume 16 (1974), pp. 83-100 | MR | Zbl

[23] Varopoulos On a commuting family of contractions on a Hilbert space, Rev. Roum. Math. Pures Appl., Volume 21 (1976) no. 9, pp. 1283-1285 | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT